AP Physics 1 & 2 Math Review

Although this is a science lesson, math will be half of what you learn in this course on top of the physics concepts so be sure to have mastered your algebra for this course. Luckily, the math in this physics course won’t be as challenging as that in your mathematics classes, however some topics like trigonometry will be used ad nauseam, so it’s best to be ready for the challenges ahead.

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Math Review

The SI system is a system of units based on the metric system designed to standardized measurements across the world.

The fundamental units of the SI system are the meter, kilogram, second, ampere, candela, Kelvin, and mole.

Significant figures represent a manner of showing which digits in a number are known to some level of certainty.

Scientific notation provides an efficient way to describing very large and very small numbers.

Accuracy is how close a measurement is to the actual value. Precision describes the repeatability of a measurement.

Math Review

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Transcription: Math Review

Hi folks, and welcome back to educator.com. What I'd like to do now, is take a few minutes to go through a review of some of the math skills we are going to need to be successful in this course.0000

In our outline, we are going to talk about the metric system and the system international, or SI units, which is the unit system that we use in physics.0010

We will talk about significant figures, scientific notation, and finally, the difference between accuracy and precision, and why they are so important.0019

The objectives are: convert and estimate SI units, recognize fundamental and derived units, express numeric quantities with correct significant figures so we understand how accurate and how precise our measurements are going to be.0028

We will use scientific notations to express physical values efficiently, and finally, differentiate between accuracy and precision.0043

So, why do we need units? Well physics involves the study of prediction and analysis of real world events and real world events have quantifiable numbers.0052

In order to communicate these to other people accurately, we need to have some sort of standards. Whether it be a sound was this loud, or this quiet. We need to put a number on that so we can communicate to people. The light was this bright, or this dim.0062

How do we put numbers around that? We have to decide on a set of standards and physicists have agreed to use what is known as the system international, which is a subset of the metric system.0080

You will also sometimes see it referred to as the MKS system because the basic units include meters, kilograms, and for time, seconds0090

Let's talk about it. The system international is comprised of seven fundamental units. It is based on powers of 10 because it is a subset of the metric system and all other units are derived from these basic seven.0109

The fundamental units are the meters, the kilograms, the second, hence the MKS system, the ampere, the candela, kelvin and the mole, which you may be familiar with from chemistry.0122

So let's start with the meter. The meter is a measure of length similar to the yard in the English system. For measurements smaller than the meter, use a centimeter which is about the width of your pinky finger perhaps. A millimeter is 1/10 of that, micrometer which is often times written μm, and nanometer, nm.0137

For measurements larger than a meter, typically we use kilometers, kilometers, 1000 meters.0158

The kilogram on the other hand, is roughly equivalent to 2.2 English pounds. For measurements smaller than a kilogram, we often times use grams or milligrams. A gram is about a paperclip.0166

For measurements larger than a kilogram, we could use things like a megagram, also known as a metric ton. That is 1000 kilograms.0179

In time, and everyone is probably familiar with this one, the base unit of time is a second. And unlike the rest of the metric system, time is a little funny. It is not based on units of 10. We have, instead, things like minutes, which is 60 seconds. Hours, which is 60 minutes. Days, which is 24 hours, and years, 365¼ days. But most of us are so familiar with this, it is not really a big deal.0189

For shorter times, we go back to base 10. For example, things like milliseconds, microseconds, and nanoseconds and so on...0212

We can take and we can make other units from these fundamental units. A unit of velocity or speed, for example is a meters per second, or if we take that further, could be a kilometer per hour. In the English system it might be a mile per hour.0219

Acceleration is a meter/second2 which is really just a meter per second every second.0237

Force is measured in newtons. But a newton is really just a kilogram times a meter divided by a second, divided by a second. That is kg×m/s2.0244

These are derived units. They are comprised of combinations of those seven fundamental units.0253

As we talk about the metric system and these powers of ten, we need to look at the prefixes.0260

If we talk about something like a kilogram, a kilogram gets the symbol k in front of the g, for gram, kilogram would be 103 grams.0267

A gigagram would be 109 grams. Micrometer would be 10-6 meters, and this table is awfully helpful for converting units.0276

Let's talk about how we convert fundamental units. If we have something like 2,480 meters and we want to convert it to kilometers, here is a nice and easy way to convert these.0282

Even if you can do it in your head, it is probably pretty good to learn this method because later on, when the units get more complicated, it will still work out for you.0302

Let's start off with what we have right now. 2,480m, and I am going to write that as a fraction so it is 2480/1.0310

I want the meters to go away, so I am going to multiply by something where I have meters in the denominator on the right hand side.0320

The units that I want are kilometers. To fill in the rest of this, what I have to realize, is that I can multiply anything by 1 and I get the same value.0328

If I multiply 3,280 by 1, I get 3,280. If I multiply 6 pigs by one, I get 6 pigs. The trick is, I can write 1 in a bunch of different ways.0341

I could write 1 as 0.5/0.5, that is equal to 1. I could write 1 as 3 apples/3 apples, that is still equal to 1.0353

So, I am going to use this math trick and I am going to multiply this by 1, but I am going to pick how I write 1 very carefully.0363

To do this, what I am going to do, is, I am trying to convert to kilometers, k. So I go over to my table of prefixes and I find k for kilo.0370

I see that it means 103, so I am going to write 103 over here on the bottom because on the bottom, there is no prefix in front of the unit.0380

If I put 103 here, I am going to put 1 on the other side. What I have now made is a ratio 1km/103m and 1km is 1000× - 103m.0392

What I have really written here is 1 but I've written it in a special way so when I multiply this through, my meters make a ratio of 1. 2,480×1km/103 is going to leave me with 2.48 and my units that are left are kilometers.0405

2,480m is 2.48km. It is a nice, simple way of converting units. Let's try another one.0428

5.357kg. Let's convert that to grams. I start by writing what I have. 5.357kg, and I write it as a ratio over 1, 5.357kg/1 ×,I want kg to go away so I will write kg in the denominator and I want grams in the numerator.0439

Now, I go to my prefix table and look up kilo, k, again is 103. I am going to write that on this side that does not have a prefix. So that goes on the top this time and put a 1 on the bottom.0460

Now kg and kg make a ratio of 1, or cancel out. What I'm left with is 5.357×103g/1. So 5.357×103;is just going to be 5,357 grams.0476

There we go, converting fundamental units. Let's take a look at a 2 step conversion. Sometimes you have to do this in a couple of different steps.0499

We want to convert 6.4×10-6milliseconds to nanoseconds. I start by writing what we have. 6.4×10-6ms/1. I want ms to go away.0509

I put ms on the bottom and I will convert to my base unit, seconds on the top. I look up what milli, m, means and it means 10-3. Again, I write that on the side that does not have a prefix.0526

So 10-3 up there and 1 on the other side. Milliseconds would make a ratio of 1 and we are left with seconds but I do not want just seconds. I want nanoseconds, so I need to do another step.0541

Multiply by, I want seconds to go away, so I will put that in the denominator and I want units of nanoseconds.0556

Now I go look up nano, n, 10-9. That again goes on the side without a prefix. I put a 1 on the other side and when I go look back here, seconds are going to cancel out.0562

When I multiply this through, 6.4×10-6×10-3/10-9 and the units I'm left with should be nanoseconds. I come up with 6.4 nanoseconds.0578

Let's go back the other way just to verify we have got this down. We already know what the answer should be here because we just did this problem, just in the other direction. Let's verify that it works.0604

6.4ns/1 and we are going to multiply. We want nanoseconds to go away so we are going to put that on the bottom and we'll go to seconds.0616

I look up nano, which means 10-9 so I write 10-9 over here on the side that does not have a prefix. I put 1 on the other side.0628

Now I'm left with seconds, but I want milliseconds so I do it again. If I want seconds to go away, I want milliseconds so I go to my table and look up milli which is 10-3.0640

It goes on the side without a prefix, I put a 1 on the other side, and as I look here, nanoseconds cancel out, seconds will cancel out, and I should be left with milliseconds.0654

So I multiply through. 6.4×10-9/10-3 gives me 6.4×10-6 and the units I'm left with are milliseconds.0666

There is my answer. It is exactly as we expected. Let's do one with some derived units.0682

We have 32m/s and we want to convert that to something like kilometers per hour. We are going to follow the same basic path again. We are going to write 32m/s as a fraction and if I want to convert to kilometers per hour, I can convert either the meters or seconds first, it does not really matter.0690

Let's start by converting the meters into kilometers. I want meters to go away, so that goes into the denominator and I want kilometers in the numerator.0710

I go to my handy dandy table over here and find that kilo means 103. That goes on the side without a prefix and 1 goes on the other side.0719

Now I'm going to be left with kilometers per second but I want kilometers per hour. So I have another step. The seconds here in the denominator, I need those to go away so I put seconds up here and it would be nice to put hours down here but I do not really know how many seconds are in an hour, but I know how many seconds are in a minute.0730

So I will do this first. I will say that there are 60 seconds in 1 minute. Now when I look at my units, my seconds will cancel out and I'm down to kilometers per minute.0750

I had best do another step here. So if I want minutes to go away, I will put that in the numerator. I want hours and I know that there are 60 minutes in 1 hour. I check my units again and minutes make a ratio of 1 and what I should be left with for units is going to be kilometers in the numerator per hour.0761

I am all set to go do my math. 32×60×60/103should give me about 115.2 kilometers per hour. A derived unit conversion problem.0782

Let's take a look at a multi-step conversion. One last unit conversion problem. Let's see how many seconds are in one year. I have no idea but it is kind of a fun problem to take a look at.0804

Let's start with 1 year, we will make that as a ratio. I do not know how many seconds are in a year but what I do know is that there are 365¼ days in 1 year. Years make a ratio of 1 and I am left with units of days.0816

We are still not to seconds but what I happen to know is if the days go away, there are 24 hours in 1 day. Days will make a ratio of one and I am down to hours. We are still not to seconds.0836

So in another step, I want hours to go away so I will convert to minutes. I know there are 60 minutes in 1 hour. Hours will make a ratio of 1 and I am down to minutes. We are getting closer.0853

I want minutes to go away so I will put minutes in the denominator. I want seconds and there are 60 seconds in 1 minute. Minutes will make a ratio of 1 and I am left with my units of seconds.0867

When I go through and I do all of this math, 1×365¼×24×60×60, I come out with about 3.16×107seconds. That is a lot of seconds in 1 year.0880

Another useful tool or skill is being able to estimate some of these units. For example, estimate the length of a football field. Well that is pretty big but just a rough ballpark figure is maybe about 100 meters.0901

If you are familiar with the English system, 100 yards and 100 meters are roughly the same thing. Or the mass of a student is maybe 60-70kg for a typical student.0919

The length of a marathon is somewhere in the ballpark of about 40, 42km or the mass of a paperclip, I think we mentioned this one previously is somewhere in the ballpark of about one gram.0931

So as you walk around and see different objects see if you can take an estimate of what their mass, their length, their time is in various units. It is a useful skill.0946

Let's talk about significant figures. Significant figures represent the manner of showing which digits in a number you know with some level of certainty.0959

For example, If you are walking along and see a garden gnome in someone's yard, significant figures can help you understand to what exactness you know the height of that garden gnome.0970

14cm, 14.3827482cm, or 14.0cm? These three numbers are all telling you slightly different things. What do they mean? Well, the key to significant figures is following these rules: Write down as many digits as you can with absolute certainty.0982

Once you have done that, go to one more decimal place, one more level of accuracy and try to take your best guess. The resulting value is your quantity in significant figures.1003

Now reading the significant figures, you start with the value in scientific notation and we will talk about that here very shortly. All non zero digits are significant. All digits that are in-between non zero digits are significant.1015

Zeros to the left of significant digits are not significant but zeros to the right of significant digits are significant.1031

As an example, how many significant digits are in the value 43.74km? Well we have 1,2,3,4 non zero digits so we must have 4 significant figures. We know at for certainty to 43.7 and that 4 is our best guess on the next level of accuracy.1040

How many significant figures are in the value of 4,302.5 grams? Well we have 4 non-zero digits and zeros between significant figures are significant so we have a total of 5 significant figures.1062

How many significant figures are in the value of .0083s? Well those are significant but zeros to the left of significant figures are not significant so here we have 2 significant figures.1081

How many significant figures are in the value 1.200×103kg? Zeros to the right of significant figures are significant so we have 1,2,3,4 significant figures.1094

Having gone through this, let's talk now about scientific notation. The need for scientific notation has to do with the tremendous variation in units, in magnitudes of these units, and their sizes.1111

For example, when we talk about length, we could talk about something like the width of a country, like the United States, which is probably a pretty big number, but we also have to talk about the thickness of human hair, all with the same base measurement of meters.1126

Even smaller, how about the transistor on the integrated circuit. Those are getting so small, it is smaller than a wavelength of light. So small that there is no optical microscope in the world that can ever see some of those features.1142

Huge ranges in orders of magnitude for these different measurements. Scientific notation can helps us express these efficiently and make it much easier to read.1154

For example, which of these numbers is easier to read. 4000000000000 or 4×1012. That is obvious, that is a lot easier to read and there is much less chance of making a mistake.1167

Or, which is easier here .0000000001m or 1×10-9m? I think it's easy to see that those are a lot more accurate and less error prone. It is almost tough to read these numbers with all of the zeros because it's so easy to lose your place in them.1182

So, using scientific notation. First off, show your value using the correct number of significant figures. Then, move the decimal point so that one significant figure is to the left of the decimal point.1200

Finally, show your number being multiplied by 10 to the appropriate power so that you get the same quantity, the same numerical value.1214

And finally let's talk about accuracy and precision. There is a difference between these two and in everyday speech, we often times use them interchangeably but in the world of physics, the world of science, There is an important distinction.1223

Accuracy is how close a measurement is to the target value. Precision, on the other hand, is how repeatable your measurements are. I like to look at these from the metaphor of target practice with a bow and arrow.1236

If we are aiming over here towards our first target and we are kind of all over the place here with our arrows and, by the way, they are nowhere close to the target and nowhere near each other, we have low accuracy and low precision which is typically not what you are after.1248

Over here, however, we have pretty high accuracy, we are starting to get close to the target but we are still not repeatable. We are accurate, close to the target but not repeatable therefore we have high accuracy and low precision.1262

Over here we are nowhere close to the target but we can hit that same spot nowhere close to the target every time. We are extremely precise, but our accuracy is off. High precision and low accuracy.1277

Finally, the nirvana of measurement, we have high accuracy, we are very near the target and we are repeatable, we have high precision. We can get near the target and we can get near the target every time.1291

With that, let's take a look at a couple more examples. Let's show this number 300,000,000 in terms of scientific notation assuming we know 3 significant figures.1303

We will find that 3 significant figures and I want to show this in scientific notation, I have one digit, one number to the left of the decimal place and I know 2 more significant figures so I write that as 3.00 to give me my 3 significant figures and I multiply it by 10 to the appropriate power which would be 1,2,3,4,5,6,7,8. 3.00×108.1316

How about showing this number, .000000... There is no way I can read this whole thing... 282 in scientific notation. Well, we have 3 significant figures so this must be 2.82×10 to some power. What power is that going to be? Well we have to move the decimal place 15 places to the right. So it would be 10-15. Isn't that a lot more efficient and easier to read?1343

How about here? Express the number .000470 in scientific notation. We have 3 significant figures, so 4.70×, and the power is going to be, 1,2,3,4 to the right, so 10-4.1387

And one last one, let's see if we can expand 1.11×107. We have 1.11 and we need to move the decimal place 7, so 1,2,3,4,5,6,7. So I would write that as 11,100,000. 11 million, 100 thousand.1408

Hopefully this gets you a good start on some of the basic math skills we are going to need here in physics especially around scientific notation, significant figures, units, converting units, and accuracy and precision. Thanks for watching educator.com, we will see you next time and make it a great day!1434

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